This invention relates to modulation methods for wireless signal transmission. More particularly, the invention relates to modulation methods that reduce the error rates of received signals in fading environments and that enable data rates to be increased without the need to increase bandwidth or transmitted power. Still more particularly, the invention relates to such methods in conjunction with the use of multiple antenna arrays.
It is generally desirable to reduce error rates, and to increase transmission rates, in wireless transmission systems. Multiple-antenna arrays can be used to achieve these desirable effects.
Fading is one of several physical phenomena that tend to increase error rates, or to reduce channel capacity, in wireless transmission systems. Fading is the result of destructive interference, at the receiver, between correlated signal portions that because of scattering have arrived over different-length paths.
One technique that tends to mitigate the effects of fading is differential phase modulation, in which phase differences carry transmitted information. Although differential phase modulation is a known technique for single-antenna transmission and reception in fading environments, there are no known adaptations of this technique for use with multiple-antenna arrays.
However, in certain fading environments, the theoretical capacity of a multiple-antenna communication link increases linearly with the size of the transmitter or receiver array, this effect being determined by the array having the lesser number of antennas. This effect has been predicted for rich scattering environments in which fading is xe2x80x9cflat.xe2x80x9d That is, the propagation coefficients that describe the effect of the physical transmission channel on the transmitted signal are approximately independent of frequency over the signal bandwidth. Flat fading can be achieved in practice for a particular environment if the bandwidth is not too great, or if it is restricted appropriately.
Significantly, such a linear increase in capacity occurs only if the propagation coefficients between all pairs of transmitter and receiver antennas are known to the receiver. In practice, this condition can be met only if the receiver is trained, from time to time, by receiving known training signals from the transmitter.
Communication methods that use such a training procedure are described, for example, in the co-pending U.S. patent application Ser. No. 08/938,168, commonly assigned herewith, filed on Sep. 26, 1997 by B. M. Hochwald et al. under the title, xe2x80x9cMultiple Antenna Communication System and Method Thereof.xe2x80x9d
Other co-pending patent applications, commonly assigned herewith, that describe related subject matter are Ser. No. 08/673,981, filed on Jul. 1, 1996 by G. J. Foschini under the title xe2x80x9cWireless Communications System Having a Layered Space-Time Architecture Employing Multi-Element Antennas,xe2x80x9d Ser. No. 09/060,657, filed on Apr. 15, 1998 by G. J. Foschini and G. D. Golden under the title xe2x80x9cWireless Communications System Having a Space-Time Architecture Employing Multi-Element Antennas at Both the Transmitter and Receiver,xe2x80x9d and a patent application filed on Jul. 10, 1998 by T. L. Marzetta under the title xe2x80x9cDetermining Channel Characteristics in a Space-Time Architecture Wireless Communication System Having Multi-Element Antennas.xe2x80x9d
Unfortunately, training intervals cut into the available time during which data may be transmitted. The length of this interval increases as the number of transmitter antennas is increased. Moreover, the propagation coefficients can be treated as constant only over an average period of time referred to as the fading coherence interval. To be effective, training should be repeated at least once per such interval. However, fading is very rapid in some environments, such as those in which a mobile station is operating within a rapidly moving vehicle. For rapid fading environments, the time between fades may be too short for the communication system to learn the propagation coefficients belonging to even one transmitting antenna, much less those of a multiple antenna array.
Thus, there remains a need to more fully realize, in practice, the theoretical benefits of multiple antenna arrays in fading environments.
In the co-pending U.S. patent application Ser. No. 09/134,297, commonly assigned herewith, filed on Aug. 14, 1998 by B. M. Hochwald et al. under the title, xe2x80x9cWireless Transmission Method for Antenna Arrays, Having Improved Resistance to Fading,xe2x80x9d there was described a new method of signal modulation. This new method, which we refer to as xe2x80x9cUnitary Space-Time Modulation (USTM),xe2x80x9d is robust against fading and receiver-induced noise in flat fading environments. Significantly, it does not require knowledge of the propagation coefficients, although in some implementations, such knowledge can be used to further improve performance.
In USTM, each message to be transmitted is transformed into a sequence of signals selected from a constellation of L possible signals, L a positive integer. (Thus, each transmitted signal embodies a number of bits given by log L. In the present discussion, xe2x80x9clogxe2x80x9d will denote the binary logarithm.) Each of these symbols is, itself, a time sequence of complex amplitudes for transmission by the transmitting antenna or antennas. (We will speak, in general terms, of a transmitting array having a plural transmitting antennas. However, it should be noted that the number M of transmitting antennas may be 1.) The transmissions by all of the antennas in the transmitting array are concerted. All of these transmissions (for a given signal) are made in the same sequence of T successive time units (which we refer to as symbol intervals), T a positive integer.
Thus, a signal may be represented by a complex-valued matrix having T rows and M columns. Each column corresponds to a respective antenna of the transmitting array, and represents the sequence of complex amplitudes to be transmitted by that antenna. Each row corresponds to a particular one of the T symbol intervals, and describes the complex amplitude to be transmitted by each respective antenna during that interval. Such a set of complex amplitudes is referred to as a xe2x80x9csymbol.xe2x80x9d Each symbol is distributed in space (i.e., across the transmitting array), and each signal is composed of T symbols distributed in time.
Significantly, each signal matrix must have the property that all of its columns are orthonormal. (It should be noted in this regard that corresponding to a signal matrix "PHgr", the baseband signals provided to the transmitting array are represented by matrix S, where S={square root over (TP)} "PHgr". Here, P is the average power fed into each antenna.) Because each of these columns has length T, there can never be more than T such columns that are all orthogonal to each other.
There are L signals, and M columns per signal. Thus, over the entire constellation, there are Lxc3x97M columns. Because there will typically be many signals in the constellation (constellation sizes in the hundreds of thousands, or even more, are desirable in at least some applications), Lxc3x97M will typically be much greater than T. Well known mathematical properties dictate that there can be no more than T mutually orthonormal column vectors. Therefore, it will be unlikely that, given a randomly chosen pair of signal matrices, the columns of one such matrix will be orthogonal to the columns of the other.
If such orthogonality between the respective columns of signal pairs were possible, the probability of confusing one received signal for another would be reduced to its ideal minimum value. Given that this ideal condition is unattainable, it is desirable, instead, to design the signal constellation in such a way that correlations between pairs of signal matrices, of a kind that tends to increase the error probability, are made as small as possible.
U.S. patent application Ser. No. 09/134,297, cited above, describes techniques for minimizing these correlations that are most useful when the number M of transmitting antennas is relatively small. What has been lacking, until now, is a more powerful technique that can readily generate signal constellations of low correlation when M, L, and T are relatively large, without demanding impractical amounts of computational resources.
Such a solution is described here.
When a signal matrix (of dimensionality Txc3x97M is left-multiplied by a Txc3x97T unitary matrix "THgr", the product is a new Txc3x97M matrix, which also has the column-orthonormality properties that qualify it to serve as a signal matrix. In general, the new Txc3x97M matrix can again be left-multiplied by "THgr", and so on ad infinitem, to generate many Txc3x97M matrices having orthonormal columns.
We have discovered that the matrix "THgr" can be tailored in such a way that the resulting product matrices tend to have relatively low correlations with each other. Thus, an appropriate subset of these product matrices is advantageously employed as a signal constellation.
A particular class of Txc3x97T unitary matrices that we have found useful in this regard is the class of matrices that are ""th roots of the Txc3x97T identity matrix. It will be appreciated that if "THgr" belongs to such a class, no more than  distinct signal matrices can be generated from an initial signal matrix by repeated applications of "THgr". The parameter  may be chosen to equal the size L of the desired signal constellation, or it may be chosen to be larger than L. If  is larger than L, the constellation is assembled by selecting a subset of size L from the generated matrices.
One useful calculation of a pairwise correlation between signal matrices "PHgr"l and "PHgr"lxe2x80x2 involves taking the product "PHgr"lxe2x80xa0"PHgr"lxe2x80x2, where the superscript xe2x80x9cxe2x80xa0xe2x80x9d denotes the conjugate transpose. We refer to this product as the correlation matrix.
The indices l and lxe2x80x2 are used to denote particular signal matrices. Thus, each of these indices takes on integer values from 1 to L. Suppose now that the initial signal matrix is indexed by 1, the result of one multiplication by "THgr" is indexed by 2, and so on. Thus, the signal matrix "PHgr"l is the result of lxe2x88x921 multiplications by "THgr". Then if "THgr" belongs to the class of ""th roots of the identity matrix, the correlation matrices will have the following property: Given indices xcex1, xcex2, xcex3, if [xcex1xe2x88x92xcex2] mod =[xcex3xe2x88x92xcex4] mod , then "PHgr"xcex1xe2x80xa0"PHgr"xcex2="PHgr"xcex3xe2x80xa0"PHgr"xcex4. When this property holds, we say that the signal matrices have a circulant correlation structure.
Accordingly, the invention in one aspect involves a method for creating a constellation of signals for wireless transmission. The method comprises providing an initial signal in the form of a complex-valued matrix, all columns of which are orthonormal. The method further comprises generating a further plurality of matrices by a process that assures that each of the generated matrices is related to the initial matrix as a product of one or more multiplications of the initial matrix by a unitary matrix.
In another aspect, the invention involves a method for the wireless transmission of signals. According to such method, at least one baseband signal is generated, and then it is transmitted, on a radio-frequency carrier, from an array of one or more transmission antennas. Each transmission of a baseband signal is carried out by transmitting a sequence of amplitudes from each of the one or more antennas of the array.
Each such transmission is made in accordance with a signal matrix having columns and rows, in which each column represents a distinct antenna of the array, and each row represents a distinct time interval, such that the entries in each column are proportional to the amplitudes to be transmitted, in sequence, from a corresponding antenna of the array. The columns of each signal matrix are orthonormal.
Each signal matrix is selected from a constellation of available signal matrices. Each matrix in the constellation is either an initial matrix, or it is related to the initial matrix as a product of one or more multiplications of the initial matrix by a unitary matrix.
In alternate embodiments of the invention, the matrices of the signal constellation are related to the initial matrix as products of multiplications of the initial matrix by one or more unitary matrices.
In particular embodiments of the invention, the signal constellation is a subset of a set of matrices having a circulant correlation structure.